Question

For a positive integer $n$, let $S\left(n\right)=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+.....+\frac{1}{2^n}$. Then $S_{100}>100$.

• A. True
• B. False

Answer 1

The correct option is B False

$a=1,r=\frac{1}{2}$

$S=\frac{a(1-r^n)}{1-r}=\frac{1-\{1-{\left(\frac{1}{2}\right)}^n\}}{1-\frac{1}{2}}=2(1-\frac{1}{2^n})$

$\therefore$ From the above we can see when $n=100,$

$S_{100}<<<100$.